CBSE Class 12 Question Paper 2024 Applied Mathematics

CBSE

666

Series &RQPS Set – 4
àíZ -nÌ H$moS>
*RQPS* Q.P. Code 465
AZwH«$_m§§H$
narjmWu àíZ-nÌ H$moS> >H$mo CÎma-nwpñVH$m Ho$
Roll No. _wI-n¥ð >na Adí` {bIo§ &
Candidates must write the Q.P. Code
on the title page of the answer-book .

· H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _o§ _w{ÐV n¥ð> 27 h¢ &
· H¥$n`m Om±M H$a b| {H$ Bg àíZ-nÌ _| > 38 àíZ h¢ &
· àíZ-nÌ _| Xm{hZo hmW H$s Amoa {XE JE àíZ-nÌ H$moS> H$mo narjmWu CÎma-nwpñVH$m Ho$
_wI-n¥ð> na {bI| &
· H¥$n`m àíZ H$m CÎma {bIZm ewê$ H$aZo go nhbo, CÎma-nwpñVH$m _| àíZ H$m H«$_m§H$
Adí` {bI| &
· Bg àíZ-nÌ H$mo n‹T>Zo Ho$ {bE 15 {_ZQ >H$m g_` {X`m J`m h¡ & àíZ-nÌ H$m {dVaU
nydm©• _| 10.15 ~Oo {H$`m OmEJm & 10.15 ~Oo go 10.30 ~Oo VH$ N>mÌ Ho$db àíZ-nÌ
H$mo n‹T>|Jo Am¡a Bg Ad{Y Ho$ Xm¡amZ do CÎma-nwpñVH$m na H$moB© CÎma Zht {bI|Jo &
· Please check that this question paper contains 27 printed pages.
· Please check that this question paper contains 38 questions.
· Q.P. Code given on the right hand side of the question paper should be
written on the title page of the answer-book by the candidate.
· Please write down the serial number of the question in the answer-book
before attempting it.
· 15 minute time has been allotted to read this question paper. The question
paper will be distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m.,
the students will read the question paper only and will not write any
answer on the answer-book during this period.

ì`mdhm[aH$ J{UV
APPLIED MATHEMATICS

{ZYm©[aV g_` : 3 KÊQ>o A{YH$V_ A§H$ : 80
Time allowed : 3 hours Maximum Marks : 80

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gm_mÝ` {ZX}e :
{ZåZ{b{IV {ZX}em| H$mo ~hþV gmdYmZr go n{‹T>E Am¡a CZH$m g™Vr go nmbZ H$s{OE :
(i) Bg àíZ-nÌ _| 38 àíZ h¢ & g^r àíZ A{Zdm`© h¢ &
(ii) `h àíZ-nÌ nm±M IÊS>m| _| {d^m{OV h¡ – H$, I, J, K Ed§ L> &
(iii) IÊS> H$ _| àíZ g§»`m 1 go 18 VH$ ~hþ{dH$ënr` VWm àíZ g§»`m 19 Ed§ 20 A{^H$WZ
Ed§ VH©$ AmYm[aV 1 A§H$ Ho$ àíZ h¢ &
(iv) IÊS> I _| àíZ g§ »`m 21 go 25 VH$ A{V bKw-CÎmar` (VSA) àH$ma Ho$ 2 A§H$m| Ho$ àíZ h¢ &

(v) IÊS> J _| àíZ g§»`m 26 go 31 VH$ bKw-CÎmar` (SA) àH$ma Ho$ 3 A§H$m| Ho$ àíZ h¢ &
(vi) IÊS> K _| àíZ g§»`m 32 go 35 VH$ XrK© -CÎmar` (LA) àH$ma Ho$ 5 A§H$m| Ho$ àíZ h¢ &

(vii) IÊS> L> _| àíZ g§»`m 36 go 38 àH$aU AÜ``Z AmYm[aV 4 A§H$m| Ho$ àíZ h¢ &

(viii) àíZ-nÌ _| g_J« {dH$ën Zht {X`m J`m h¡ & `Ú{n, IÊS> I Ho$ 2 àíZm| _|, IÊS> J Ho$ 2 àíZm|
_|, IÊS> K Ho$ 2 àíZm| _| VWm IÊS> L> Ho$ 3 àíZm| _| Am§V[aH$ {dH$ën H$m àmdYmZ {X`m J`m
h¡ &
(ix) H¡$ëHw$boQ>a H$m Cn`moJ d{O©V h¡ &
IÊS> H$

Bg IÊS> _| ~hþ{dH$ënr` àíZ h¢, {OZ_| àË`oH$ àíZ 1 A§H$ H$m h¡ &

1. 1 km H$s Xm¡‹S> _|, {Ibm‹ S>r P, {Ibm‹S>r Q H$mo 18 _rQ>a `m 9 goH§$S> go ham XoVm h¡ & Xm¡‹S
nyar H$aZo Ho$ {bE P H$m g_` Š`m h¡ ?
(A) 512 goH§$S> (B) 502 goH§$S>

(C) 491 goH§$S> (D) 481 goH§$S>

2. `{X x > y VWm z < 0 hmo‚ Vmo :
(A) xz > yz (B) xz ³ yz
x y x y
(C) > (D) <
z z z z

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General Instructions :
Read the following instructions very carefully and strictly follow them :
(i) This question paper contains 38 questions. All questions are compulsory.
(ii) This question paper is divided into five Sections – A, B, C, D and E.
(iii) In Section A, Questions no. 1 to 18 are multiple choice questions (MCQs) and
questions number 19 and 20 are Assertion-Reason based questions of 1 mark
each.
(iv) In Section B, Questions no. 21 to 25 are very short answer (VSA) type
questions, carrying 2 marks each.
(v) In Section C, Questions no. 26 to 31 are short answer (SA) type questions,
carrying 3 marks each.
(vi) In Section D, Questions no. 32 to 35 are long answer (LA) type questions
carrying 5 marks each.
(vii) In Section E, Questions no. 36 to 38 are case study based questions carrying
4 marks each.
(viii) There is no overall choice. However, an internal choice has been provided in
2 questions in Section B, 2 questions in Section C, 2 questions in Section D and
3 questions in Section E.
(ix) Use of calculators is not allowed.

SECTION A

This section comprises multiple choice questions of 1 mark each.

1. In a 1 km race, player P beats player Q by 18 metres or 9 seconds. What
is P’s time to complete the race ?

(A) 512 seconds (B) 502 seconds

(C) 491 seconds (D) 481 seconds

2. If x > y and z < 0, then :

(A) xz > yz (B) xz ³ yz
x y x y
(C) > (D) <
z z z z
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3. `{X AB = A Am¡a BA = B hmo‚ Vmo (B2 + B) ~am~a h¡ :
(A) 2A (B) O
(C) 2I (D) 2B

42 2 5
4. ∆ = 79 7 9 H$m _mZ h¡ :
29 5 3

(A) 0 (B) 1
(C) –3 (D) –15

3
5. `{X y = e–2x h¡‚ Vmo d y3 ~am~a h¡ :
dx

(A) 2 e –2x (B) e –4x

(C) 4 e –4x (D) – 8 e –2x

6. \$bZ f(x) = x2 - x + 1 h¡ :
(A) (0, 1) _| dY©_mZ
(B) (0, 1) _| õmg_mZ
1 1
(C) (0, ) _| dY©_mZ Am¡a ( , 1) _| õmg_mZ
2 2
1 1
(D) ( , 1) _| dY©_mZ Am¡a (0, ) _| õmg_mZ
2 2

7. AdH$b g_rH$aU
æyö
y dx + x log ç ÷ dy - 2x dy = 0
èxø
H$s H$mo{Q> Ed§ KmV H« $_e: h¢ :
(A) 1, 1 (B) 1, 2
(C) 2, 1 (D) 1, n[a^m{fV Zht
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3. If AB = A and BA = B, then (B2 + B) equals :

(A) 2A (B) O

(C) 2I (D) 2B

42 2 5
4. The value of ∆ = 79 7 9 is :
29 5 3

(A) 0 (B) 1

(C) –3 (D) –15

–2x d3 y
5. If y = e , then is equal to :
dx 3

(A) 2 e –2x (B) e –4x

(C) 4 e –4x (D) – 8 e –2x

6. The function f(x) = x2 – x + 1 is :
(A) increasing in (0, 1)
(B) decreasing in (0, 1)
1 1
(C) increasing in (0, ) and decreasing in ( , 1)
2 2
1 1
(D) increasing in ( , 1) and decreasing in (0, )
2 2

7. The order and the degree of the differential equation
æyö
y dx + x log ç ÷ dy - 2x dy = 0
èxø

are respectively :

(A) 1, 1 (B) 1, 2

(C) 2, 1 (D) 1, not defined

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8. EH$ Ý`m` {gŠH$m Xmo ~ma CN>mbm OmVm h¡ Am¡a n[aUm_ ZmoQ> H$a {b`m OmVm h¡ & `{X Bg
narjU _| {MVm| H$s g§»`m H$mo Xem©Zo dmbm `mÑpÀN>H$ Ma X hmo‚ Vmo X H$s J{UVr` àË`mem
(expectation) hmoJr :
1
(A) 1 (B)
2
1 1
(C) (D) 1
4 2

9. `{X dV©_mZ g_`, am{Ì 9:00 ~Oo h¢, Vmo 1275 K§Q>m| Ho$ ~mX H$m¡Z-gm g_` hmoJm ?
(A) am{Ì 11 ~Oo (B) am{Ì 12 ~Oo
(C) am{Ì 9 ~Oo (D) gw~h 9 ~Oo

10. `{X EH$ ßdmgm| Ma X Ho$ {bE,
P(X = k) = P(X = k + 1) h¡‚

Vmo X H$m àgaU hmoJm :
(A) k–1 (B) k
(C) k+1 (D) k+2

11. `{X |t| H$m n[aH${bV _mZ tv(a) (t H$m H«$m§{VH$ _mZ) go H$_ hmo‚ Vmo {ZamH$aUr`
n[aH$ënZm :
(A) AñdrH$ma H$s OmVr h¡ &
(B) ñdrH$ma H$s OmVr h¡ &
(C) Z Vmo ñdrH$ma H$s OmVr h¡ Am¡a Z hr AñdrH$ma &
(D) {ZYm©[aV Zht H$s Om gH$Vr h¡ &

12. Xmo ñdV§Ì Z_yZm| Ho$ _mÜ`m| Ho$ ~rM Ho$ A§Va H$s gmW©H$Vm H$s Om±M H$aZo Ho$ {bE, ñdmV§Í`
H$mo{Q> (v) ____________ br OmVr h¡ &
(A) n1 – n2 + 2 (B) n1 – n2 – 2

(C) n1 + n2 – 2 (D) n1 + n2 + 2

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8. A fair coin is tossed twice and outcomes are noted. If the random variable
X represents the number of heads that appeared in the experiment, then
the mathematical expectation of X is :
1
(A) 1 (B)
2
1 1
(C) (D) 1
4 2

9. What time will it be after 1275 hours, if the present time is 9:00 p.m. ?
(A) 11 p.m. (B) 12 p.m.
(C) 9 p.m. (D) 9 a.m.

10. If for a Poisson variate X,

P(X = k) = P(X = k + 1),

then the variance of X is :
(A) k–1 (B) k
(C) k+1 (D) k+2

11. If the calculated value of |t|< tv (a) (critical value of t), then the null
hypothesis :
(A) is rejected.
(B) is accepted.
(C) is neither accepted nor rejected.
(D) cannot be determined.

12. For testing the significance of difference between the means of two
independent samples, the degree of freedom (v) is taken as :
(A) n1 – n2 + 2 (B) n1 – n2 – 2

(C) n1 + n2 – 2 (D) n1 + n2 + 2

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13. {XE JE _mZm| 23, 32, 40, 47, 58, 33, 42 Ho$ {bE, 5-dfu` J{V_mZ Am¡gV h¢ :
(A) 38, 40, 42 (B) 40, 42, 44

(C) 40, 42, 46 (D) 42, 44, 46

14. âb¡Q> aoQ> nÕ{V H$m Cn`moJ H$aVo hþE, 8% dm{f©H$ ã`mO Xa na 2 1 dfm] _| < 20,000 H$m
2
G$U MwH$mZo Ho$ {bE EMI hmoJm :
(A) < 700 (B) < 800
(C) < 900 (D) < 100

15. EH$ _mo~mBb \$moZ H$m _yë` < 12,000 h¡ Am¡a 3 df© Ho$ Cn`moJr OrdZ Ho$ ~mX‚ CgH$m
ñH¡«$n _yë` < 3,000 h¡ & Vmo 2 df© Ho$ A§V _| _mo~mBb \$moZ H$m ~wH$ _yë` hmoJm :
(A) < 3,000 (B) < 6,000

(C) < 5,000 (D) < 7,000

16. 8 dfm] _| < 50,000 O_m H$aZo Ho$ {bE àË`o H$ 6 _hrZo Ho$ A§ V _| {H$VZr YZam{e O_m H$s
OmZr Mm{hE‚ `{X YZam{e na ã`mO 6% dm{f©H$ h¡ Am¡a ã`mO AY©-dm{f©H$ g§`mo{OV hmoVm
h¡ ? [{X`m J`m h¡ : (1·03)16= 1·6047]
(A) < 3,432·53 (B) < 2,783·08
(C) < 2,480·57 (D) < 2,149·93

17. Ag_rH$aU 2x + 3y > 6 H$m AmboI h¡ :
(A) nyU© XOY-Vb
(B) AY©-Vb {Og_| _yb-q~Xþ pñWV h¢
(C) AY©-Vb {Og_| Z Vmo _yb-q~Xþ Am¡a Z hr aoIm 2x + 3y = 6 Ho$ q~Xþ pñWV h¢
(D) nyU© XOY-Vb {Og_| aoIm 2x + 3y = 6 Ho$ q~Xþ em{_b Zht h¢
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13. For the given values 23, 32, 40, 47, 58, 33, 42; the 5-yearly moving
averages are :
(A) 38, 40, 42 (B) 40, 42, 44
(C) 40, 42, 46 (D) 42, 44, 46

1
14. Using flat rate method, the EMI to repay a loan of < 20,000 in 2 years
2
at an interest rate of 8% p.a. is :
(A) < 700 (B) < 800
(C) < 900 (D) < 100

15. A mobile phone costs < 12,000 and its scrap value after a useful life of
3 years is < 3,000. Then, the book value of the mobile phone at the end
of 2 years is :
(A) < 3,000 (B) < 6,000
(C) < 5,000 (D) < 7,000

16. What sum of money should be deposited at the end of every 6 months to
accumulate < 50,000 in 8 years, if money is worth 6% p.a. compounded
16
semi-annually ? [Given : (1·03) = 1·6047]
(A) < 3,432·53 (B) < 2,783·08
(C) < 2,480·57 (D) < 2,149·93

17. The graph of the inequation 2x + 3y > 6 is the :
(A) entire XOY-plane
(B) half-plane that contains the origin
(C) half-plane that neither contains the origin nor the points on the
line 2x + 3y = 6
(D) whole XOY-plane excluding the points on the line 2x + 3y = 6

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18. EH$ a¡{IH$ àmoJ«m_Z g_ñ`m _|‚ `{X CÔoí` \$bZ Z = ax + by H$m A{YH$V_ _mZ gwg§JV
joÌ Ho$ Xmo H$moZr` q~XþAm| na EH$g_mZ h¡ ‚ Vmo q~XþAm| H$s g§»`m {OZ na Z H$m A{YH$V_
_mZ h¡, h¡ :
(A) 0 (B) 2

(C) gr{_V (D) AZ§V

àíZ g§»`m 19 Am¡a 20 A{^H$WZ Ed§ VH©$ AmYm[aV àíZ h¢ & Xmo H$WZ {XE JE h¢ {OZ_| EH$ H$mo
A{^H$WZ (A) VWm Xÿgao H$mo VH©$ (R) Ûmam A§{H$V {H$`m J`m h¡ & BZ àíZm| Ho$ ghr CÎma ZrMo {XE
JE H$moS>m| (A), (B), (C) Am¡a (D) _| go MwZH$a Xr{OE &
(A) A{^H$WZ (A) Am¡a VH©$ (R) XmoZm| ghr h¢ Am¡a VH©$ (R), A{^H$WZ (A) H$s ghr
ì¶m»¶m H$aVm h¡ &
(B) A{^H$WZ (A) Am¡a VH©$ (R) XmoZm| ghr h¢, naÝVw VH©$ (R), A{^H$WZ (A) H$s ghr
ì¶m»¶m Zht H$aVm h¡ &
(C) A{^H$WZ (A) ghr h¡, naÝVw VH©$ (R) Jµ bV h¡ &
(D) A{^H$WZ (A) µJbV h¡, naÝVw VH©$ (R) ghr h¡ &

19. A{^H$WZ (A) : \$bZ f(x) = x2 – x + 1, A§Vamb (– 1, 1) na {Za§Va dY©_mZ h¡ &
VH©$ (R) : `{X f(x) A§Vamb [a, b] na g§VV Am¡a A§Vamb (a, b) na AdH$bZr`
h¡‚ Vmo A§Vamb [a, b] _| f(x) {Za§Va dY©_mZ h¡, `{X g^r xÎ (a, b) Ho$
{bE f¢(x) > 0 h¡ &

20. EH$ {ÛnX ~§Q>Z _|, n = 200 Am¡a p = 0·04 h¡ & ßdmgm| ~§Q>Z H$mo {ÛnX ~§Q>Z Ho$ g{ÞH$Q>Z
Ho$ ê$n _| boVo hþE :

A{^H$WZ (A) : ßdmgm| ~§Q>Z H$m _mÜ` 8 h¡ &
512
VH©$ (R) : P (X = 4) = .
3e8

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18. In an LPP, if the objective function Z = ax + by has same maximum value
on two corner points of the feasible region, then the number of points at
which maximum value of Z occurs is :

(A) 0 (B) 2

(C) finite (D) infinite

Questions number 19 and 20 are Assertion and Reason based questions. Two
statements are given, one labelled Assertion (A) and the other labelled Reason
(R). Select the correct answer from the codes (A), (B), (C) and (D) as given below.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the
correct explanation of the Assertion (A).

(B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not
the correct explanation of the Assertion (A).

(C) Assertion (A) is true, but Reason (R) is false.

(D) Assertion (A) is false, but Reason (R) is true.

19. Assertion (A) : The function f(x) = x 2 – x + 1 is strictly increasing on
(– 1, 1).

Reason (R) : If f(x) is continuous on [a, b] and derivable on (a, b), then
f(x) is strictly increasing on [a, b] if f¢(x) > 0 for all
x Î (a, b).

20. In a binomial distribution, n = 200 and p = 0·04. Taking Poisson
distribution as an approximation to the binomial distribution :

Assertion (A) : Mean of Poisson distribution = 8.
512
Reason (R) : P (X = 4) = .
3e8

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IÊS> I

Bg IÊS> _| A{V bKw-CÎmar` (VSA) àH$ma Ho$ àíZ h¢, {OZ_| àË`oH$ Ho$ 2 A§H$ h¢ &

é 1 0ù
21. (H$) `{X A = ê -1 7 ú h¡‚ Vmo k H$m dh _mZ kmV H$s{OE {OgHo$ {bE
ë û
2
A – 8A + kI = 0 h¡ &

AWdm

é x - y 2x + z ù é -1 5 ù
(I) `{X ê 2x - y 3z + w ú = ê 0 13ú h¡‚ Vmo x, y, z Am¡a w Ho$ _mZ kmV
ë û ë û
H$s{OE &

22. H¡«$_a {Z`_ H$m Cn`moJ H$aHo$‚ {ZåZ{b{IV g_rH$aU {ZH$m` H$mo hb H$s{OE :

2x1 + 3x2 = 5

11x1 – 5x2 = 6

23. {ZåZ{b{IV a¡{IH$ àmoJ«m_Z g_ñ`m H$m AmboIr` {d{Y go hb kmV H$s{OE (`{X H$moB© hb
h¡ Vmo) :
ì`damoYm|
x–y£–1
–x+y£0
x, y ³ 0
Ho$ A§VJ©V
Z = x + y H$m A{YH$V_rH$aU H$s{OE &

24. 6% dm{f©H$, {Ogo Ì¡_m{gH$ g§`mo{OV {H$`m J`m h¡ na‚ àË`oH$ {V_mhr Ho$ A§V _| Xo`
< 600 H$s emídVVm H$m dV©_mZ _yë` kmV H$s{OE &

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SECTION B

This section comprises very short answer (VSA) type questions of 2 marks each.

é 1 0ù
21. (a) If A = ê ú , find the value of k such that A2 – 8A + kI = 0.
ë -1 7 û

OR

é x - y 2x + z ù é -1 5 ù
(b) If ê ú= ê ú , find the values of x, y, z and w.
ë2x - y 3z + w û ë 0 13û

22. Using Cramer’s rule, solve the following system of equations :

2x1 + 3x2 = 5

11x1 – 5x2 = 6

23. Find the solution to the following linear programming problem (if it
exists) graphically :

Maximize Z = x + y
subject to the constraints

x–y£–1
–x+y£0
x, y ³ 0.

24. At 6% p.a., compounded quarterly, find the present value of a perpetuity
of < 600 payable at the end of each quarter.

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25. (H$) _mZ br{OE {H$ {H$gr {Zdoe H$m ewéAmVr _yë` < 20,000 h¡ Am¡a `h 3 df© _|
~‹T>H$a < 50,000 hmo OmVm h¡ & CAGR (MH«$d¥{ÕV dm{f©H$ d¥{Õ Xa) H$s JUZm
H$s{OE & [br{OE (2·5)1/3 = 1·355]
AWdm
(I) EH$ ì`{º$ EH$ dñVw < 12,000 _| IarXVm h¡ & df© Ho$ A§V _|‚ dh Bg dñVw H$mo
< 15,000 _| ~oM XoVm h¡ & `{X _wÐmñ\$s{V H$s Xa 6% Wr‚ Vmo [aQ>Z© H$s Zm__mÌ
Am¡a dmñV{dH$ Xa| kmV H$s{OE &

IÊS> J
Bg IÊS> _| bKw-CÎmar` (SA) àH$ma Ho$ àíZ h¢, {OZ_| àË`oH$ Ho$ 3A§H$ h¢ &

26. EH$ H§$Q>oZa _| 50 brQ>a Oyg h¡ & Bg_| go 5 brQ>a Oyg {ZH$mb {b`m OmVm h¡ Am¡a H§$Q>oZa _|
5 brQ>a nmZr S>mb {X`m OmVm h¡ & Bg {H«$`m H$mo 4 ~ma Am¡a Xmoham`m OmVm h¡ & Bg {H«$ `m
Ho$ AÝV _| H§$Q>oZa _| Oyg H$s _mÌm kmV H$s{OE & [br{OE (0·9)5 = 0·59049]

2
27. (H$)
ò
0
x 2 dx H$m _mZ kmV H$s{OE, AV: J«m\$ na dh joÌ Xem©BE {OgH$mo `h

joÌ\$b {Zê${nV H$aVm h¡ &
AWdm
1
e–x
(I) _mZ kmV H$s{OE :
ò 1 + e dx
0
x

28. àW_ MVwWmªe _| ~Zo Eogo d¥Îmm| Ho$ Hw$b H$m AdH$b g_rH$aU kmV H$s{OE Omo XmoZm|
{ZX}em§H$ Ajm| H$mo ñne© H$aVo h¢ &

29. `h {X`m J`m h¡ {H$ EH$ IQ narjU _| Cå_rXdmam| Ho$ EH$ g_yh Ho$ A§H$ gm_mÝ` ~§{Q>V hmoVo
h¢ & `{X IQ narjU H$m _mÜ` 100 Am¡a _mZH$ {dMbZ 10 h¡‚ Vmo àm[`H$Vm kmV H$s{OE
{H$ narjm XoZo dmbm Cå_rXdma 90 Am¡a 110 Ho$ ~rM ñH$moa H$aoJm &
[{X`m J`m h¡ : P (Z < 1) = 0·8413 Am¡a P (Z < – 1) = 0·1587]

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25. (a) Assume an investment’s starting value is < 20,000 and it grows to
< 50,000 in 3 years. Calculate CAGR (Compounded Annual
Growth Rate) [Use : (2·5)1/3 = 1·355]
OR
(b) A man bought an item for < 12,000. At the end of the year, he
decided to sell it for < 15,000. If the inflation rate was 6%, find the
nominal and real rate of return.

SECTION C

This section comprises short answer (SA) type questions of 3 marks each.

26. A container has 50 litres of juice in it. 5 litres of juice is taken out and is
replaced by 5 litres of water. This process is repeated 4 more times.
Determine the quantity of juice in the container after final replacement.
[Use (0·9)5 = 0·59049]
2
27. (a) Evaluate :
ò
0
x 2 dx and hence show the region on the graph whose

area it represents.
OR
1
e– x
(b) Evaluate :
ò 1 + e dx
0
x

28. Find the differential equation of all circles in the first quadrant which
touches both the coordinate axes.

29. Given that the scores of a set of candidates on an IQ test are normally
distributed. If the IQ test has a mean of 100 and a standard deviation of
10, determine the probability that a candidate who takes the test will
score between 90 and 110.

[Given P (Z < 1) = 0·8413 and P (Z < – 1) = 0·1587]

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30. 20 EO|{g`m| _| 4-n{h`m dmhZ H$s Am¡gV gmßVm{hH$ {~H«$s 50 `y{ZQ> à{V EO|gr Wr & EH$
{dkmnZ A{^`mZ Ho$ ~mX‚ Am¡gV gmßVm{hH$ {~«H$s ~‹T>H$a 55 `y{ZQ> à{V EO|gr
10 `y{ZQ> _mZH$ {dMbZ g{hV hmo JB© Wr & Om±M H$s{OE {H$ Š`m `h {dkmnZ A{^`mZ
g\$b ahm Wm & [{X`m J`m h¡ : 5 = 2 × 24, t19(0 × 05) = 1 × 729]

31. (H$) hmb hr _| EH$ boIm§H$Z ñZmVH$ Zo EH$ Z`m ì`dgm` Imobm Am¡a EH$ H§$ß`yQ>a
{gñQ>_ ñWm{nV {H$`m {OgH$s bmJV < 45,200 Wr & H§$ß`yQ>a {gñQ>_ H$m _yë`õmg
a¡{IH$ ê$n go 3 dfm] Ho$ {bE hmoJm Am¡a BgH$m ñH«$¡ n _yë` < 0 hmoJm &
(i) _yë`õmg H$s Xa Š`m h¡ ?
(ii) EH$ a¡{IH$ g_rH$aU kmV H$s{OE Omo t df© Ho$ A§V _| H§$ß`yQ>a {gñQ>_ Ho$
~wH$ d¡ë`y H$m dU©Z H$aVm h¡‚ Ohm± 0 £ t £ 3 h¡ &
(iii) S>o‹T> df© Ho$ A§V _| H§$ß`yQ>a {gñQ>_ H$s ~wH$ d¡ë`y Š`m hmoJr ?
AWdm
(I) à^mdr Xa kmV H$s{OE Omo 10% à{V df© H$s gm_mÝ` Xa Ho$ ~am~a h¡, Omo …
(i) AY©-dm{f©H$ g§`moo{OV hmoVr h¡ &
(ii) {V_mhr g§`mo{OV hmoVr h¡ &
[{X`m J`m h¡ : (1·05)2 = 1·1025, (1·025)4 = 1·1038]

IÊS> K

Bg IÊS> _| XrK©-CÎmar` (LA) àH$ma Ho$ àíZ h¢, {OZ_| àË`oH$ Ho$ 5 A§H$ h¢ &

32. EH$ Q>§H$s _| VrZ nmBn A, B Am¡a C bJo h¢ & nmBn A Am¡a B BZboQ> nmBn h¢ O~{H$ nmBn
C EH$ AmCQ>boQ> nmBn h¡ & nmBn A Am¡a B Q>§H$s H$mo H«$_e: 3 K§ Q>o Am¡a 4 K§ Q>o _| ^a XoVo h¢
O~{H$ nmBn C nyar ^ar Q>§H$s H$mo 1 K§Q>o _| nyar Vah go Imbr H$a XoVm h¡ & `{X `o VrZm|
nmBn A, B Am¡a C H«$_ go gw~h 5, 6 VWm 7 ~Oo Imobo OmVo h¢, Vmo nyar Q>§H$s {H$VZo g_`
_| Imbr hmo OmEJr ?

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666
30. The mean weekly sales of a 4-wheeler was 50 units per agency in
20 agencies. After an advertising campaign, the mean weekly sales
increased to 55 units per agency with standard deviation of 10 units. Test
whether the advertising campaign was successful.
[Given 5 = 2 × 24, t19(0 × 05) = 1 × 729]

31. (a) A recent accounting graduate opened a new business and installed
a computer system that costs < 45,200. The computer system will
be depreciated linearly over 3 years and will have a scrap value of
< 0.
(i) What is the rate of depreciation ?
(ii) Give a linear equation that describes the computer system’s
book value at the end of tth year, where 0 £ t £ 3.
(iii) What will be the computer system’s book value at the end of
the first year and a half ?
OR
(b) Find the effective rate which is equivalent to normal rate of 10%
p.a. compounded :
(i) semi-annually.
(ii) quarterly.
[Given (1·05)2 = 1·1025, (1·025)4 = 1·1038]

SECTION D

This section comprises long answer (LA) type questions of 5 marks each.

32. A cistern has three pipes A, B and C. Pipes A and B are inlet pipes
whereas C is an outlet pipe. Pipes A and B can fill the cistern separately
in 3 hours and 4 hours respectively; while pipe C can empty the
completely filled cistern in 1 hour. If the pipes A, B and C are opened in
order at 5, 6 and 7 a.m. respectively, at what time will the cistern be
empty ?

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3 4 45 2
33. (H$) f(x) = – x – 8x3 – x + 105 Ûmam àXÎm \$bZ Ho$ {bE ñWmZr` CƒV_
4 2
Am¡a ñWmZr` {ZåZV_ Ho$ g^r q~XþAm| H$mo kmV H$s{OE &
AWdm
(I) A§Vamb kmV H$s{OE {OZ_| f(x) = 20 - 9x + 6x2 - x3 Ûmam àXÎm \$bZ f {Za§Va
dY©_mZ `m {Za§Va õmg_mZ h¡ &

34. (H$) _mZ br{OE X EH$ `mÑpÀN>H$ ê$n go M`{ZV ñHy$b {Xdg Ho$ Xm¡amZ ~mahdt H$jm
Ho$ N>mÌ Ûmam AÜ``Z {H$E JE K§Q>m| H$s g§»`m Xem©Vm h¡ & àm{`H$Vm `h h¡ {H$
AkmV pñWam§H$ ‘k’ Ho$ {bE X Ho$ _mZ xi hmo gH$Vo h¢ :
ì 0·1 `{X xi = 0
ï
P(X = k) = í kx i `{X x i = 1 `m 2
ïk(5 – x ) `{X x i = 3 `m 4
î i

(i) k H$m _mZ kmV H$s{OE &
(ii) N>mÌ Ho$ H$_-go-H$_ 2 K§Q>o AÜ``Z H$aZo H$s àm{`H$Vm kmV H$s{OE &
(iii) N>mÌ Ho$ A{YH$-go-A{YH$ 2 K§Q>o AÜ``Z H$aZo H$s àm{`H$Vm kmV H$s{OE &
AWdm
(I) EH$ N>moQ>o eha Ho$ nmg EH$ ZXr _| ha 10 gmb _| Xmo ~ma ~m‹T> AmVr h¡ Am¡a dh
Amodaâbmo hmo OmVr h¡ & `h _mZVo hþE {H$ ßdmgm| ~§Q>Z C{MV h¡, _mÜ` Anojm Š`m
h¡ ? gmW hr 10 df© Ho$ A§Vamb _| 3 `m Cggo H$_ Amodaâbmo VWm ~m‹T> H$s
àm{`H$Vm kmV H$s{OE &
[{X`m J`m h¡ : e–2 = 0·13534]

35. A_¥Vm EH$ H$ma IarXVr h¡ {OgHo$ {bE dh < 2,50,000 H$m A{J«_ ^wJVmZ H$aVr h¡ Am¡a
eof am{e H$m ^wJVmZ 2 dfm] _| < 25,448 àË`oH$ H$s _m{gH$ {H$íV Ûmam {H$`m OmZm h¡ &
`{X \$mBZ|ga 20% n«{V df© H$s Xa go ã`mO boVm h¡‚ Vmo H$ma H$s dmñV{dH$ H$s_V kmV
-24
H$s{OE & [{X`m J`m h¡ : æç 61 ÷ö = 0 × 67253]
è 60 ø
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666
33. (a) Find all the points of local maxima and local minima of the
function :

3 4 45 2
f(x) = – x – 8x3 – x + 105.
4 2
OR
(b) Find the intervals in which the following function f is strictly
increasing or strictly decreasing :
f(x) = 20 - 9x + 6x2 - x3 .

34. (a) Let X denote the number of hours a Class 12 student studies
during a randomly selected school day. The probability that X can
take the values xi, for an unknown constant ‘k’ :
ì 0·1 if xi = 0
ï
P(X = k) = í kx i if x i = 1 or 2
ïk(5 – x ) if x i = 3 or 4
î i

(i) Find the value of k.
(ii) Determine the probability that the student studied for at
least 2 hours.
(iii) Determine the probability that the student studied for at
most 2 hours.
OR
(b) A river near a small town floods and overflows twice in every
10 years on an average. Assuming that the Poisson distribution is
appropriate, what is the mean expectation ? Also, calculate the
probability of 3 or less overflows and floods in a 10-year interval.
[Given e–2 = 0·13534]

35. Amrita buys a car for which she makes a down payment of < 2,50,000

and the balance is to be paid in 2 years by monthly instalments of

< 25,448 each. If the financer charges interest at the rate of 20% p.a, find
-24
æ 61 ö
the actual price of the car. [Given ç ÷ = 0·67253]
è 60 ø
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IÊS> L>

Bg IÊS> _| 3 àH$aU AÜ``Z-AmYm[aV àíZ h¢ {OZ_| àË`oH$ Ho$ 4 A§H$ h¢ &
àH$aU AÜ``Z – 1

36. AnZo OÝ_{XZ na‚ ào_m Zo AZmWmb` Ho$ ~ƒm| H$mo Hw$N> n¡go XmZ H$aZo H$m \¡$gbm {H$`m &

`{X 8 ~ƒo H$_ hmoVo h¢, Vmo àË`oH$ ~ƒo H$mo < 10 A{YH$ àmßV hmoVo h¢ & naÝVw, `{X 16
~ƒo A{YH$ hmoVo h¢, Vmo àË`oH$ ~ƒo H$mo < 10 H$_ àmßV hmoVo h¢ &
_mZm AZmWmb` _| ~ƒm| H$s g§»`m x VWm àË`oH$ ~ƒo H$mo XmZ go {_bZo dmbr am{e
< y h¡ &

Cn`w©º$ gyMZm Ho$ AmYma na, {ZåZ{b{IV àíZm| Ho$ CÎma Xr{OE :

(i) Xr JB© pñW{V go ~Zo x Am¡a y _| a¡{IH$ g_rH$aU {ZH$m` {b{IE & 1

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SECTION E

This section comprises 3 case study-based questions of 4 marks each.

Case Study – 1

36. On her birthday, Prema decides to donate some money to children of an
orphanage home.

If there are 8 children less, everyone gets < 10 more. However, if there
are 16 children more, everyone gets < 10 less.
Let the number of children in the orphanage home be x and the amount
to be donated to each child be < y.

Based on the above information, answer the following questions :

(i) Write the system of linear equations in x and y formed of the given
situation. 1

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(ii) D$na (i) _| {bIo JE a¡{IH$ g_rH$aU {ZH$m` H$mo Amì`yh g_rH$aU AX = B Ho$
ê$n _| {b{IE & 1

(iii) (H$) Amì`yh A H$m à{Vbmo_ kmV H$s{OE & 2
AWdm
(I) x Am¡a y Ho $ _mZ kmV H$s{OE & 2

àH$aU AÜ``Z – 2
37. g§»`m {gÕm§V _|‚ nyUmªH$ N Ho$ JwUZI§S> ImoOZm AH$ga _hÎdnyU© hmoVm h¡ & g§»`m N Ho$
Xmo VwÀN> JwÊ mZI§S> h¢‚ AWm©V 1 Am¡a N & `{X H$moB© AÝ` JwUZI§S> _m¡OyX h¡‚ Vmo Cgo N H$m
µJ¡a-VwÀN> JwUZI§S> H$hm OmVm h¡ & Zaoe Zo q~XþAm| A (0, 50), B (20, 40), C (50, 100),
D (0, 200) Am¡a E (100, 0) Ho $ gmW Hw$N> ì`damoYm| (a¡{IH$ Ag_mZVmAm|) H$m EH$ J«m\$
V¡`ma {H$`m & `h J«m\$ VrZ µJ¡a-VwÀN> ì`damoYm| Am¡a Xmo VwÀN> ì`damoYm| go ~Zm h¡ & BZ VrZ
µJ¡a-VwÀN> ì`damoYm| _| EH$ ì`damoY h¡ x + 2y ³ 100.

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(ii) Write the system of linear equations, obtained in (i) above, in
matrix form AX = B. 1

(iii) (a) Find the inverse of matrix A. 2
OR
(b) Determine the values of x and y. 2

Case Study – 2

37. In number theory, it is often important to find factors of an integer N.
The number N has two trivial factors, namely 1 and N. Any other factor,
if exists, is called non-trivial factor of N. Naresh has plotted a graph
of some constraints (linear inequations) with points A (0, 50), B (20, 40),
C (50, 100), D (0, 200) and E (100, 0). This graph is constructed using three
non-trivial constraints and two trivial constraints. One of the non-trivial
constraints is x + 2y ³ 100.

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Cn`w©º$ gyMZm Ho$ AmYma na, {ZåZ{b{IV àíZm| Ho$ CÎma Xr{OE :

(i) Xmo VwÀN> ì`damoY H$m¡Z -H$m¡Z go h¢ ? 1

(ii) (H$) `{X gwg§JV joÌ R1 h¡‚ Vmo AÝ` Xmo µJ¡a-VwÀN> ì`damoY Š`m h¢ ? 2

AWdm

(I) `{X gwg§JV joÌ R2 h¡‚ Vmo AÝ` Xmo µJ¡a-VwÀN> ì`damoY Š`m h¢ ? 2

(iii) `{X gwg§JV joÌ R1 h¡‚ Vmo CÔoí` \$bZ z = 5x + 2y H$m A{YH$V_ _mZ kmV
H$s{OE & 1

àH$aU AÜ``Z – 3

38. O~ b§~o g_` VH$ XoIm OmVm h¡, Vmo EH$ g_`-ûm¥§Ibm Am±H$‹S>o CZ éPmZm| H$s ^{dî`dmUr
H$a gH$Vo h¢ Omo {dMmamYrZ Ma _| d¥{Õ `m H$_r `m R>hamd H$m AZw_mZ bJm gH$Vo h¢ & Bg
Vah Ho$ {díbofUmË_H$ AÜ``Z {H$gr ì`dgm` H$mo ^{dî` _| AZw_m{ZV {~H«$s `m CËnmXZ
H$s ^{dî`dmUr `m nydm©Zw_mZ Ho$ {bE bm^Xm`H$ hmo gH$Vm h¡ &
{ZåZ{b{IV Vm{bH$m 1996 – 2001 Ho$ Xm¡amZ EH$ {Obo _| EH$ dñVw H$s {~H«$s H$mo Xem©Vr
h¡ …

df© : 1996 1997 1998 1999 2000 2001

{~H«$s (< bmI _|) : 6·5 5·3 4·3 6·1 5·6 7·8

Cn`wº© $ gyMZm Ho$ AmYma na, {ZåZ{b{IV àíZm| Ho$ CÎma Xr{OE :

(i) grYr-aoIm àd¥{Îm H$m g_rH$aU {ZYm©[aV H$s{OE & 2

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Based on the above information, answer the following questions :

(i) What are the two trivial constraints ? 1

(ii) (a) If R1 is the feasible region, then what are the other two
non-trivial constraints ? 2

OR

(b) If R2 is the feasible region, then what are the other two
non-trivial constraints ? 2

(iii) If R1 is the feasible region, then find the maximum value of the
objective function z = 5x + 2y. 1

Case Study – 3

38. When observed over a long period of time, a time series data can predict
trends that can forecast increase or decrease or stagnation of a variable
under consideration. Such analytical studies can benefit a business for
forecasting or prediction of future estimated sales or production.

The table below shows the sale of an item in a district during
1996 – 2001 :

Year : 1996 1997 1998 1999 2000 2001

Sales (in lakh <) : 6·5 5·3 4·3 6·1 5·6 7·8

Based on the above information, answer the following questions :

(i) Determine the equation of the straight-line trend. 2

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(ii) (H$) dfm] Ho$ éPmZ _yë`m| H$mo gmaUr~Õ H$s{OE Am¡a df© 2002 Ho$ {bE Ano{jV
{~H«$s àd¥{Îm H$s ^r JUZm H$s{OE & 2

AWdm

(I) {ZåZ{b{IV Am±H$‹S>m| Ho$ {bE Ý`yZV_ dJ© {d{Y Ûmam EH$ grYr-aoIm àd¥{Îm
H$mo {\$Q> H$s{OE … 2
df© : 2004 2005 2006 2007 2008 2009 2010

bm^ (< ’000) 114 130 126 144 138 156 164

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(ii) (a) Tabulate the trend values of the years and also compute
expected sales trend for the year 2002. 2

OR

(b) Fit a straight-line trend by the method of least squares for
the following data : 2

Year : 2004 2005 2006 2007 2008 2009 2010

Profit (< ’000) 114 130 126 144 138 156 164

465-11 Page 27 of 27 P.T.O.

CBSE STUDY MATERIAL

CBSE Board

INFORMATION CLASS STUDY MATERIAL

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